nLab
Gelfand spectrum

Idea

The Gelfand spectrum of a commutative C-star algebra $A$ is a topological space $X$ such that $A$ is the algebra of complex-valued continuous functions on $X$. (This “Gelfand duality” is a special case of the general duality between spaces and their algebras of functions.)

Definition

Given a unital, not necessarily commutative, complex $C^{*}$-algebra $A$, the set of its continuous characters, that is continuous nonzero linear homomorphisms into the field of complex numbers, is canonically equipped with a so-called spectral topology which is compact Hausdorff. (If applied to a nonunital $C^{*}$-algebra, then it is only locally compact.) This correspondence extends to a functor, called the Gel’fand spectrum (also Gelfand, originally Гельфанд) from the category of unital $C^{*}$-algebras to the category of Hausdorff spaces, which is full and faithful when restricted to the subcategory of commutative unital $C^{*}$-algebras. The kernel of a character is clearly a codimension-$1$ closed subspace, and in particular a closed maximal ideal in $A$; therefore the Gel’fand spectrum is a topologised analogue of the maximal spectrum of a discrete algebra.

There is some redundancy, because a character on a unital Banach algebra is automatically continuous (with Lipschitz constant 1). I wasn’t sure if I should remove the word or if it is there for emphasis.

For noncommutative $C^{*}$-algebras the spaces of equivalence classes of irreducible representations (i.e., the spectrum) and their kernels (i.e., the primitive ideal space) are more important than the character space. Should that be mentioned here? The Gelfand spectrum is also useful in the context of more general commutative Banach algebras.

-Jonas Meyer

Related concepts

• Isbell duality

  • Gelfand duality

    • C-star algebra, topological space

    • Gelfand spectrum

    • constructive Gelfand duality theorem

category: analysis